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\section*{\usemenu{slac-pub-7131::context::slac-pub-7131-0-0-0u2}{$\hbeta$ as the LSP}}\label{section::slac-pub-7131-0-0-0u2}
In order to proceed with a discussion about the dark matter qualities
of the $\hbeta$ LSP, we must discuss its composition and mass.
For convention purposes we write down the neutralino
mass matrix
\begin{equation}
\left(
\begin{array}{cccc}
M_1 & 0 & -M_Z \cos\beta\sin\theta_W & M_Z\sin\beta\sin\theta_W \\
0 & M_2 & M_Z\cos\beta\cos\theta_W & -M_Z\sin\beta\cos\theta_W \\
-M_Z\cos\beta\sin\theta_W & M_Z\cos\beta\cos\theta_W & 0 & -\mu \\
M_Z\sin\beta\sin\theta_W & -M_Z\sin\beta\cos\theta_W & -\mu & 0
\end{array}
\right)
\end{equation}
in the $\left\{ \tilde B,\tilde W^3,-i\tilde H_d^0,
-i\tilde H^0_u\right\}$ basis.
If $0< -\mu < (M_1\simeq M_2),$ and $\tan\beta$ is near one, then
the two
lightest eigenstates of the neutralino mass matrix are $N_2 \sim \tilde
\gamma$ (photino-like), and $N_1 \sim
\sin\beta \tilde H_d^0 + \cos\beta \tilde H_u^0
+\delta \tilde Z$ (Higgsino-like), where $\delta < 0.1$.
This arrangement of lightest neutralino mass eigenstates enhances the
important radiative neutralino decay $N_2 \to \hbeta\gamma$, and along
with the $ee\gamma\gamma+\slashchar{E}_T$ event of ref.~\cite{3}
implies $m_{N_2}-m_{\hbeta}\gsim 30\gev$ and
$30 \lsim m_\hbeta \lsim 55 \gev$~\cite{5}. We
shall see that with $\tan\beta$ near one the $Z$ invisible width constraint
is satisfied and $\hbeta$ provides an interesting amount of dark matter
as well.
From the invisible width determinations at LEP, the $Z$ is not allowed
to decay into $\hbeta\hbeta$ with a partial width more than about 5 MeV
(at $2\sigma$)~\cite{7}. In our approximation, the formula for the partial
width is
\begin{equation}
\Gamma_{\mbox{inv}}=\frac{\alpha M_Z}{24 \sin^2\theta_W\cos^2\theta_W}
\cos^2 2\beta \left( 1-4\frac{m_\hbeta^2}{M_Z^2}\right)^{3/2}.
\end{equation}
Figure~\docLink{slac-pub-7131-0-0-0u2.tcx}[inv:fig]{1} has contours of the invisible width in units of MeV.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\epsfysize=3.25in
\hspace*{0in}
\epsffile{invisible.ps}
\caption{Contours of constant invisible width due to $Z\to N_1 N_1$.
The labelled lines are in units of MeV, and the current $2\sigma$
bound at LEP on the invisible width is $5\mev$.}
\label{inv:fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As can be seen from the above equation as long as $\tan\beta$ is
close enough to 1 then the invisible width constraint can be satisfied.
We think this constraint should not be applied too tightly at this stage
since it could be affected by new physics. However, since $\tan\beta \simeq 1$
is the natural region for the radiative decay requirements
in ref.~\cite{5}, this constraint does not cause
any problems for our analysis here, even if it is applied with the
most stringent assumptions~\cite{8}.
$\hbeta$ pairs annihilate through the $Z$ into fermion pairs. To
look at the prediction for $\Omega h^2$, we can expand the thermally
averaged annihilation cross section~\cite{9} into fermions $(f)$ in
the following way:
\begin{equation}
(\sigma v)(x) =\cos^2 2\beta \sum_f (a_f+b_f x)
\end{equation}
where $a_f$ and $b_f$ depend only on one unknown, the LSP mass.
Applying the usual approximation
method~\cite{10} to solve the Boltzmann equation, the relic
abundance can be found:
\begin{equation}
\Omega h^2=(2.5\times 10^{-11})\left( \frac{T_\hbeta}{T_\gamma}\right)^3
\left( \frac{T_\gamma}{2.7\,\mbox{K}}\right)^3
\frac{\sqrt{N_F}}{\cos^22\beta}
\left( \frac{\mbox{GeV}^{-2}}{ax_f+\frac{1}{2}bx_f^2}\right)
\end{equation}
where $N_F$, $(T_\hbeta/T_\gamma)^3$ and $x_f$ must be solved for
self-consistently.
Calculations such as these could be valid to a
factor of two or better.
In Fig.~\docLink{slac-pub-7131-0-0-0u2.tcx}[omega:fig]{2} contours of $\Omega h^2$ are plotted in the
$\tan\beta - m_\hbeta$ plane.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\epsfysize=3.25in
\hspace*{0in}
\epsffile{omega.ps}
\caption{Contours of constant $\Omega h^2$ for the Higgsino-like LSP
described in the text.}
\label{omega:fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Since the annihilation cross section is proportional to $\cos^2 2\beta,$
when $\tan\beta$ gets closer to 1, $\Omega h^2$ begins to exceed one.
The $t$ channel sfermion exchange is greatly suppressed (since $\hbeta$
is mainly Higgsino-like), but if the $\tilde Z$ fraction of $\hbeta$
is large enough, then a $t$ channel sfermion diagram which couples like
the $SU(2)_L$ gauge coupling could start to become important. However,
this potentially efficient annihilation channel is suppressed by a
factor of $\delta^4$ and $\delta$ is less than about
$0.1$~\cite{5}. Comparing $\delta^4/m^4_{\tilde f}$ with
$\cos^2 2\beta/M_Z^4$, we find that this channel can compete with the
$s$ channel $Z$ exchange only when $\tan\beta$ is less than about
$1.05$. Furthermore, as $\tan\beta$ gets closer and closer to 1,
$\delta$ necessarily becomes smaller and smaller, and the number
$\tan\beta <1.05$ is actually an overestimate. At $\tan\beta =1.05$ we
find that the annihilation cross section is too small for all values of
$m_\hbeta$ in Fig.~\docLink{slac-pub-7131-0-0-0u2.tcx}[omega:fig]{2}, and therefore we set this as our
lower bound on allowed $\tan\beta$ in this scenario. Consequently, the
sfermion annihilation channel is not numerically important.
The $s$ channel Higgs exchange diagrams might possibly play an important
role in the annihilation cross section. However, if the pseudo-scalar
($A^0$) is sufficiently heavy then the pseudo-scalar and heavy scalar
($H^0$) Higgses will decouple; in the absence of information about the
heavy Higgs bosons we assume they effectively decouple. In
this limit it can be shown that the $\hbeta\hbeta h^0$ also vertex decouples.
Furthermore, since the LSP is sufficiently light not to annihilate
into top quarks, or vector bosons, the light final state fermion masses
contribute a further suppression of the $h^0$ mediated annihilation
cross section.
The allowed mass range for $\hbeta$ from ref.~\cite{5}
overlaps $M_Z/2.$ If consistency with the supersymmetry interpretation
of the LEP $Z\rightarrow b\bar b$ excess is required, probably $M_\hbeta
\lsim 40 \gev$, but it is premature to assume that. If $M_\hbeta \simeq
M_Z/2$ it is necessary to do the resonant calculation very
carefully~\cite{11}. There is always a value of $\tan\beta$ for
which the curves of Fig. 2 continue across $M_Z/2$ smoothly, so we will
wait until $m_\hbeta$ and $\tan\beta$ are better measured to do the more
precise calculations needed. We show results in Fig. 2 for $M_\hbeta <
M_Z/2,$ which we expect is the most relevant region. Given the results
of reference~\cite{5},
the only channel that could complicate the simple
analysis is coannihilation of the $\hbeta$ with the
$\tilde t_1$~\cite{13}, if
$m_{\tilde t_1} \simeq m_\hbeta$ ($\tilde t_1$ is the lightest stop mass
eigenstate). We expect $m_{\tilde t_1} \gsim M_Z/2$ and $M_\hbeta \lsim
M_Z/2$, so probably this complication can be ignored, but until the
masses are better determined it should be kept in mind.
The Hubble constant $h$ is probably between about $0.5$ and $0.8$.
Assuming the cold dark matter constitutes 0.4 to 0.8 of
$\Omega_{\hbox{tot}}$, we expect that $\Omega_\hbeta h^2$ should lie
somewhere between $0.08$ and $0.5$ in Fig.~\docLink{slac-pub-7131-0-0-0u2.tcx}[omega:fig]{2} (e.g. $0.57^2
\times .75 = 0.25$). We emphasize that Fig. 2 follows from the results
of ref.~\cite{5}, and that apart from the approximations
mentioned above this is a prediction of the supersymmetric
interpretation of the CDF event. Further, we note that the
supersymmetric interpretation of the reported excess of $Z\rightarrow
b\bar b$ decays at LEP leads to the same region of parameters as
ref.~\cite{5}, with Higgsino-like $\hbeta$ and with
$\tan\beta$ near 1~\cite{12}, and therefore can conservatively be viewed
as consistent with this prediction, or optimistically as additional
evidence for its correctness.
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